116 research outputs found
Bounds and Invariant Sets for a Class of Switching Systems with Delayed-state-dependent Perturbations
We present a novel method to compute componentwise transient bounds, ultimate
bounds, and invariant regions for a class of switching continuous-time linear
systems with perturbation bounds that may depend nonlinearly on a delayed
state. The main advantage of the method is its componentwise nature, i.e. the
fact that it allows each component of the perturbation vector to have an
independent bound and that the bounds and sets obtained are also given
componentwise. This componentwise method does not employ a norm for bounding
either the perturbation or state vectors, avoids the need for scaling the
different state vector components in order to obtain useful results, and may
also reduce conservativeness in some cases. We give conditions for the derived
bounds to be of local or semi-global nature. In addition, we deal with the case
of perturbation bounds whose dependence on a delayed state is of affine form as
a particular case of nonlinear dependence for which the bounds derived are
shown to be globally valid. A sufficient condition for practical stability is
also provided. The present paper builds upon and extends to switching systems
with delayed-state-dependent perturbations previous results by the authors. In
this sense, the contribution is three-fold: the derivation of the
aforementioned extension; the elucidation of the precise relationship between
the class of switching linear systems to which the proposed method can be
applied and those that admit a common quadratic Lyapunov function (a question
that was left open in our previous work); and the derivation of a technique to
compute a common quadratic Lyapunov function for switching linear systems with
perturbations bounded componentwise by affine functions of the absolute value
of the state vector components.Comment: Submitted to Automatic
Ultimate boundedness of droop controlled Microgrids with secondary loops
In this paper we study theoretical properties of inverter-based microgrids
controlled via primary and secondary loops. Stability of these microgrids has
been the subject of a number of recent studies. Conventional approaches based
on standard hierarchical control rely on time-scale separation between primary
and secondary control loops to show local stability of equilibria. In this
paper we show that (i) frequency regulation can be ensured without assuming
time-scale separation and, (ii) ultimate boundedness of the trajectories
starting inside a region of the state space can be guaranteed under a condition
on the inverters power injection errors. The trajectory ultimate bound can be
computed by simple iterations of a nonlinear mapping and provides a certificate
of the overall performance of the controlled microgrid.Comment: 8 pages, 1 figur
Diagonal Riccati Stability and Applications
We consider the question of diagonal Riccati stability for a pair of real
matrices A, B. A necessary and sufficient condition for diagonal Riccati
stability is derived and applications of this to two distinct cases are
presented. We also describe some motivations for this question arising in the
theory of generalised Lotka-Volterra systems
Sliding modes in a class of complex-valued nonlinear systems
© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksA number of physical systems allow a complex-valued representation. This article extends the theory of sliding modes to a class of nonlinear systems described by complex-valued variables. Hence, states, parameters, control actions, and sliding manifolds belong, in general, to the complex field. In this article, it is also shown that the proposed design in the complex-valued framework provides shorter reaching times to the sliding manifold than the standard sliding-mode design at equal initial condition and control effort. Different implementation approaches are also evaluated, and numerical examples illustrate the proposal.Peer ReviewedPostprint (author's final draft
Review on computational methods for Lyapunov functions
Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function
Robust nonlinear control of vectored thrust aircraft
An interdisciplinary program in robust control for nonlinear systems with applications to a variety of engineering problems is outlined. Major emphasis will be placed on flight control, with both experimental and analytical studies. This program builds on recent new results in control theory for stability, stabilization, robust stability, robust performance, synthesis, and model reduction in a unified framework using Linear Fractional Transformations (LFT's), Linear Matrix Inequalities (LMI's), and the structured singular value micron. Most of these new advances have been accomplished by the Caltech controls group independently or in collaboration with researchers in other institutions. These recent results offer a new and remarkably unified framework for all aspects of robust control, but what is particularly important for this program is that they also have important implications for system identification and control of nonlinear systems. This combines well with Caltech's expertise in nonlinear control theory, both in geometric methods and methods for systems with constraints and saturations
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